Simulation of MA(1) Longitudinal Negative Binomial Counts and Using a Generalized Methods of Moment Equation Approach

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1 Chaot Modeln and Smulaton CMSM 3: Smulaton of MA Lontudnal Neatve Bnomal Counts and Usn a Genealzed Methods of Moment Equaton Appoah Naushad Mamode Khan epatment of Eonoms and StatstsUnvest of MautusRedut Mautus E-mal:n.mamodekhan@uom.a.mu Abstat Lontudnal ount data often ase n fnanal and medal studes. n suh applatons the data exhbt moe vaablt and thus the vaane to mean ato s eate than one. Unde suh umstanes the neatve bnomal s moe onvenent to be used fo modeln these lontudnal esponses. Sne these esponses ae olleted ove tme fo the same subjet t s moe lkel that the wll be oelated. n lteatue vaous oelaton models have been poposed and amon them the most popula ae the autoeessve and the movn aveae stutues. Besdes these esponses ae often subjet to multple ovaates that ma be tme-ndependent o tme-dependent. n the event of tme-ndependene t s elatvel eas to smulate and model the lontudnal neatve bnomal ounts follown the MA stutues but as fo the ase of tmedependene the smulaton of the MA lontudnal ount esponses s a hallenn poblem. n ths pape we wll use the bnomal thnnn opeaton to eneate sets of MA non-statona lontudnal neatve bnomal ounts and the effen of the smulaton esults ae assessed va a enealzed method of moments appoah. Kewods: Neatve Bnomal Lontudnal Movn Aveae Bnomal thnnn Statona Nonstatona Genealzed method of moments ntoduton n toda s ea lontudnal data has beome extemel useful n applatons elated to the health and fnanal setos. t onsttutes of a numbe of subjets that ae measued ove a spefed numbe of tme ponts. Sne these measuements ae olleted fo a patula subjet on a epettve bass t s moe lkel that the data wll be oelated. he oelaton stutues ma be follown autoeessve movn aveae equoelaton unstutued o an othe eneal autooelaton stutues45. Moeove n lontudnal studes the esponses ae nfluened b man fatos suh as n the analss of C4 ounts the nfluental fatos ae the teatment ae ende and man othes. n ode to estmate the ontbuton and the snfane of eah of these fatos Reeved: 8 Jul 04 / Aepted: 5 eembe CMSM SSN

2 36 Khan towads the esponse vaable t s mpotant to tansfom the data set-up nto a eesson famewok. n lteatue the eesson paametes have been estmated b vaous appoahes. ntall the method of Genealzed Estmatn equatons GEE wee developed but t fals unde msspefed oelaton stutue patulal unde the ndependene oelaton stutue 5. heeafte Pente and Zhao developed a Jont Estmaton appoah to estmate jontl the eesson and oelaton paametes and elded moe effent eesson estmates than the GEE appoah but the jont estmaton s based on hhe ode moments. he appoah s also based on the wokn oelaton stutue but the pesene of these hh ode moments dlute the msspefaton effet and boost the effen of the estmates. On the othe hand Qu and Lndsa 3 developed an adaptve quadat nfeene based Genealzed Method of Moments GMM appoah whee the assumed powes of the empal ovaane mates as the bases. hese bases ae then used to fom soe vetos o moment estmatn equatons and theeafte the wee ombned to fom a quadat funton n a smla wa as the GMM appoah. hs appoah of analzn lontudnal eesson models has so fa been tested on nomal Posson data 3 but has not et been exploed n neatve bnomal oelated ounts data. n ths pape ou objetves ae to develop the moment estmatn equatons based neatve bnomal model onstut the quadat nfeene funton and then obtan the eesson estmates b maxmzn the funton. Howeve one hallenn ssue s that sne the neatve bnomal model s a two paamete model that s dependn on the mean and ove-dspeson paamete t mples that we wll eque hhe ode moments. hs estmaton appoah wll be tested va smulatons on MA statona and non-statona neatve bnomal ounts. he oanzaton of the pape s as follows: n the next seton we wll evew the neatve bnomal model alon wth ts MA Gaussan autooelaton stutue and the adaptve GMM appoah follown Qu and Lndsa 3. n seton 3 we wll develop the estmatn equatons fo the neatve bnomal model followed b smulaton esults. Neatve Bnomal model Lontudnal data ompse of data that ae olleted epeatedl ove t 3 tme ponts fo subjets 3. hus an th andom obsevaton at t tme pont wll have a epesentaton of the fom t. he neatve bnomal model fo t s ven b t t f t t! t t E t exp x and a t 0 wth notaton fom t t t t t ~ NeBn t t th whee n

3 Chaot Modeln and Smulaton CMSM 3: ven a p veto of ovaates fom... p x t and veto of eesson paametes of the... t... and... t.... Sne these ounts t ae olleted epeatedl ove tme t s moe lkel that t wll be oelated ove tme. n ths pape we wll assume that the smulated t set of esponse vaables ome fom the faml of MA Gaussan autooelaton stutue. he devaton of the MA statona neatve bnomal ounts follows fom MKenze bnomal thnnn poess. Howeve the devaton of the MA non-statona oelaton stutue has not et appeaed n statstal lteatue. n the next seton we povde an n-depth devaton of the MA non-statona Gaussan autooelaton stutue. 3 MA Non-Statona Gaussan autooelaton Stutues n the non-statona set-up the mean paamete at eah tme pont wll dffe as the ovaates ae tme-dependent that t Follown MKenze we set up the famewok to eneate the MA non-statona Gaussan autooelaton stutue. the bnomal thnnn poess assumes that t d t t d t whee d NeBn t ~ t ~ Beta t t t = j pob b j t == t b z j t pob b =0=- t j t t and and hat s the ondtonal dstbuton of t d t follows the bnomal dstbuton wth paametes d t and t. Unde these assumptons t ~ NeBn t and the set of... t... t an be poved that follows the MA stutue.. Unde these dstbutonal assumptons we note that the ovaane

4 38 Khan between t and t k s ven b las the ovaane does not exst. tk tk fo k and fo othe 4. Smulaton of MA Non Statona NB ounts he smulaton poess wll follow fom the bnomal thnnn opeaton explaned n the pevous seton wth t exp x t that s we need to povde a ven set of ovaate desns and a set of eesson veto that espets the dmenson of the ovaate matx. Note that fo the statona ase the ovaate matx wll be tme ndependent whle fo the non-statona the ovaate desn wll be tme-dependent. As suh we assume fo the non-statona ase the follown desns esn A esn B esn C x t 0.5t bnom30. t t pos t t t... 4 x t x t 0.5t sn t t exp t t ost t... 4 t t ln t t t t... 4 and xt s eneated fom the Posson dstbuton wth mean paamete. n ths wa the mean paamete fo eah subjet wll va. hus fo these set of ovaates and ntal estmate of the eesson veto dspeson paamete and oelaton paamete we eneate MA Neatve Bnomal andom vaables b fst smulatn the eo

5 Chaot Modeln and Smulaton CMSM 3: omponents t d t and the thnnn opeaton andom vaables t t. Fo ou smulaton poess we wll assume the values of. 5. Estmaton Methodolo Qu and Lndsa 3 have developed an estmaton appoah based Genealzed Methods of Moments that do not eque an assumpton n the undeln oelaton stutue and do not eque an estmaton of the oelaton paamete. n fat Qu and Lnsda 3 assumed a soe veto that onl needs the empal ovaane estmaton matx whee s the adent matx: t t and s an othoonal veto. he alulaton of the paamete eques the onjuate adent method see Qu and Lndsa 3. n the ontext of the neatve bnomal model the soe veto s defned as: f f whee the vetos f f E f f and t whee t t 0 Usn the soe veto Qu and Lndsa 3 defned the objetve funton C Q

6 40 Khan whee C s the sample vaane of 3 B maxmzn the objetve funton wth espet to the unknown set of paametes we obtan the estmatn equaton Q wth. Sne the above estmatn equaton s non-lnea we solve the equaton usn the Newton-Raphson poedue that elds an teatve equaton of the fom ˆ ˆ ˆ ˆ Q Q C Q whee C s the double devatve hessan pat of the soe funton and ths s ben used fo alulatn the vaane of the eesson and ovedspeson paametes. As llustated b Qu and Lndsa 3 ths method elds onsstent and effent estmatos and tends towads asmptot nomalt fo lae sample sze. 6. Results and Conluson Follown the pevous setons we have un 0000 smulatons fo eah of the sample based on the dffeent ovaate desns fo the non- szes 500 statona set-ups. Note that fo the statona ase the mean s held onstant at all tme ponts whlst fo non-statona the mean vaes wth the tme ponts ven the tmedependent ovaates. he table povdes the smulated mean estmates of the eesson paametes alon wth the standad eos n bakets. esn A esn B esn C ; ;0.0.0; ; ; ; ; ; ; ; ; ; ; ;0.0.00; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;0.076 Based on the smulaton esults we note that the estmates of the eesson paametes ae lose to the populaton values and as the sample sze neases the standad eos of the eesson paametes deease whh ndates that the estmates ae onsstent and

7 Chaot Modeln and Smulaton CMSM 3: effent. Howeve we have emaked a snfant numbe of falues n the smulatons as we nease the sample sze. hese falues wee manl due to ll-ondtoned natue of the double devatve Hessan matx. o oveome ths poblem n some smulatons we have used the Mooe Penose enealzed nvese method n R nv n Lba MASS to pefom the teatve poedues. Oveall the enealzed method of moments estmaton tehnque s a statstall sound tehnque but n tems of omputaton t ma not alwas be elable. Refeenes. E. MKenze. Autoeessve movn-aveae poesses wth neatve bnomal and eomet manal dstbutons. Advaned Appled Pobablt R. Pente R. & L. Zhao 99. Estmatn equatons fo paametes n means and ovaanes of multvaate dsete and ontnuous esponses. Bomets A.Qu & B. Lndsa 003. Buldn adaptve estmatn equatons when nvese of ovaane estmaton s dffult. Jounal of Roal Statstal Soet B. Sutadha. An ovevew on eesson models fo dsete lontudnal esponses. Statstal Sene B. Sutadha B. & K. as. On the effen of eesson estmatos n enealzed lnea models fo lontudnal data. Bometka